A is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number.

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5
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4answers
135 views

how do I find the continued fraction of root n ?? [duplicate]

I saw a site where they explained it.but it required calculator.I want to do it without calculator. Can anyone please help me?
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0answers
57 views

3D extension of Euclidean algorithm jigsaw method - help!

Recently I've been learning about how the Euclidean algorithm = jigsaw method (filling a rectangle with squares) = forming continued fractions. And today I'm wondering how a 3D version of the jigsaw ...
1
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1answer
35 views

continues function statement in real analysis [closed]

I ran into a challenge, i read following sentence in one note. anyone could describe or prove it for me? F is a continues function at point $ x_0 \Leftrightarrow (x_n \to x_0 \Rightarrow ...
1
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1answer
51 views

Inequality with continued fractions: $\theta_r \geq a_{r+2}\theta_{r+1} + \theta_{r+1}$

I want to prove that the following inequality is true (or that is false, I don not know but I think it is true). $$\theta_r \geq a_{r+2}\theta_{r+1} + \theta_{r+1}.$$ Here the notation is as follow: ...
1
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1answer
59 views

Approximating $\frac{t^2}{3-\frac{t^2}{5-\frac{t^2}{7-\frac{t^2}{9-\cdots}}}}$

What is a good approximation for $$\omega=\frac{t^2}{3-\frac{t^2}{5-\frac{t^2}{7-\frac{t^2}{9-\cdots}}}}$$ This will be used to find $$T=\frac{t}{1-\omega}$$ Without using Lambert's continued fraction ...
2
votes
1answer
182 views

New mathematical constant formed by continued fraction with prime numbers?

Notational convention: $$\bigoplus_{k=0}^{\infty}a_k=a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cfrac{1}{\ddots}}}}$$ Let $$ P:=\bigoplus_{k=1}^{\infty}p_k$$ where $p_k$ is the k-th prime ...
1
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0answers
24 views

Name/properties of a difference of continuants

(This is cross-posted at http://mathoverflow.net/questions/181619/name-of-a-difference-of-continuants) Suppose that $q_1$, $\ldots$, $q_s$ is a sequence of positive integers. Denote by $[q_1, ...
1
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0answers
104 views

Sums of nested radicals

Is there a known example of an infinite sum of finitely nested radicals that evaluates to a given value? Or an infinite sum of convergents of an infinite continued fraction? The finitely nested ...
7
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0answers
182 views

Continued fraction and double series.

From Euler's continued fraction formula, we have $$x = \cfrac{1}{1 - \cfrac{r_1}{1 + r_1 - \cfrac{r_2}{1 + r_2 - \cfrac{r_3}{1 + r_3 - \ddots}}}}\,$$ and $$x = 1 + \sum_{i=1}^\infty r_1r_2\cdots r_i = ...
1
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0answers
60 views

Finding Function Representation of Recursive Sequence

I was trying to find one of the roots of $x^2 + 4x + 3 = 0$ by deriving a continued fraction from the recursive formula $x = -3/x - 4$ (every step of the approximation you increase the recursion by ...
6
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2answers
197 views

Evaluation of a class of continued fractions

Is there a closed-form way of writing the continued fraction: $$ 1 + \frac{2}{3+ \frac{4}{5 + \frac{6}{7 + ...}}} $$ EDIT: Since the above has been determined as $\frac{1}{\sqrt{e}-1}$, is there a ...
9
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1answer
200 views

Minimum of $|az_x-bz_y|$

I am trying to minimize the following function: \begin{align} &f(z_x,z_y)=|az_x-bz_y| \\ &\text{ s.t. } z_x,z_y \in \mathbb{Z},1 \le z_x \le N_x \text{ and } 1 \le z_y \le N_y \text{ and } ...
2
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1answer
91 views

Continued fraction expansion of Pi (oeis A001203). [duplicate]

I would like to understand how you get the numbers $$3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{292+...}}}}$$ i.e. $\{3,7,15,1,292,...\}$ (A001203). In the comments of A046965 is explained a method ...
6
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3answers
306 views

Where did the negative answer come from in the continued fraction $1+\frac{1}{1+1/(1+\dots)}$?

In this question we increased solution domain by squaring both sides of equation but what about this one ? Here the question is to evaluate ...
1
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1answer
82 views

Induction proof for continued fractions

Recently while preparing for a maths test, I got this question in a book: Let $a(n) = 3 + \cfrac{1}{3+\cfrac{1}{3+\cfrac{1}{3+\cdots }}}$ till $n$ terms. Prove that $a(n) \cdot a(n-1)=3a(n-1)+1$ ...
0
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1answer
36 views

Continued fraction inequality: $q_n\left|q_n\alpha-p_n\right|(a_{n+1}+1)>1$

In an article it is used the fact that $$q_n\left|q_n\alpha-p_n\right|(a_{n+1}+1)>1$$ where $\alpha=[a_0;a_1,\ldots]$ is an irrational number and $q_i$ is the series of the best approximation ...
1
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1answer
58 views

Approximating a Continued Fraction

From a paper I was reading, If: $$w=\frac {1}{3}\left\{ \frac {-\dfrac {3}{16}\lambda^2}{1}+\frac {-\dfrac {3}{16}\lambda^2}{1}+\frac {-\dfrac {3}{16}\lambda^2}{1}+\frac {-\dfrac ...
1
vote
1answer
56 views

L. Jacobsen and H. Waadeland: Glimt fra analytisk teori for kjedebrøker. Del 2.

I am trying to find the aforementioned paper online but have had no luck. I originally spotted it as a reference [26] for the paper Gauss, Landen, Ramanujan, the Arithmetic-Geometric Mean, Ellipses, ...
2
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1answer
45 views

Using continued fractions to well-approximate a quadratic form?

Continued fractions are the "best rational approximation" of other numbers. For a real number $\alpha$ the continued fraction algorithm produces a sequence of integers $\alpha = [a_1, a_2, \dots, ...
2
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0answers
64 views

Continued Fraction Expansion

While reading "Gauss, Landen, Ramanujan, the Arithmetic-Geometric Mean, Ellipses, π, and the Ladies Diary " [p.602] from $F\left( -\dfrac {1}{2},-\dfrac {1}{2};1;\lambda^{2}\right)=1+\dfrac {\lambda ...
2
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1answer
103 views

Unsure about infinite continued fraction

How do you/is it possible to express $a=\cfrac{1}{2+\cfrac{3}{4+\cfrac{5}{6+\cdots}}}$ in the form $\frac{p}{q}(k+\sqrt{n})$? I'm still in high school, so I'm not familiar with especially ...
3
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2answers
120 views

Find a number by the decimal part of its square root [duplicate]

I have a math problem consisting of two questions: can we find a number N knowing only the decimal part of its square root up to a precision (only an approximation of the decimal part because the ...
12
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1answer
374 views

How to prove this determinant is $\pi$?

prove or disprove $$\pi=\begin{vmatrix} 3&1&0&0&0&\cdots\\ -1&6&1&0&0&\cdots\\ 0&-1&\dfrac{6}{3^2}&1&0&\cdots\\ ...
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1answer
77 views

Is $\{1,1,2,3,4,5,\cdots,i,\cdots \} $ the simple continued fraction algebraic or transcendental?

Is $$1+\cfrac{1}{1+\cfrac{1}{2+\cdots}} $$ or$\{1,1,2,3,4,5,\cdots,i,\cdots \} , i\in \mathbb{N}$ the simple continued fraction algebraic or transcendental? Any reference is appreciated EDIT and ...
0
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1answer
26 views

Any upper bound for $a_i$ in $\gamma =\{a_0,a_1,\dots,a_i,\dots\}$ the simple continued fraction expansion of real positive algebraic numbers?

Are there any upper bound for $a_i$ in $\gamma =\{a_0,a_1,\dots,a_i,\dots\}$ the simple continued fraction expansion of real positive algebraic numbers?
3
votes
2answers
80 views

Different Definitions Of The Sine Function

I was hoping someone could give me a flow chart or high-level map connecting all of the definitions of the sine function, with some of the reasons why we care next to each. I've tried this but I'm not ...
2
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1answer
45 views

Great Common Division with Continued Fractions

If I have this GCD equation: $$89=16\cdot5+9\\ 16=9\cdot1+7\\ 9=7\cdot1+2\\ 7=2\cdot3+1\\ 2=1\cdot2+0$$ Then my continued fraction will be: $[5: 1, 1, 3, 2]$ But if I will have this GCD equation: ...
4
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1answer
106 views

Upward continued fractions

Has anybody seen "upward continued fractions", such as $$ \frac{1+\large{\frac{1+\large{\frac{1+...}{a_2}}}{a_1}}}{a_0} \quad? $$ These can be formed, for any real number $x$ with $0<x\le 1$, by ...
3
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0answers
139 views

Help understanding a geometric proof of the ergodicity of the Gauss measure for continued fractions

Any $x\in(0,1)$ can be written as a (regular) continued fraction $$ x = \cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cdots}}} = [a_1,a_2,a_3,\dotsc] $$ An irrational number has a unique expansion, ...
0
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0answers
112 views

Semi-convergent of continued fractions

I have read this from here The simple continued fraction for $x$ generates all of the best rational approximations for $x$ according to three rules: Truncate the continued fraction, and ...
2
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1answer
64 views

Decomposition of modular group elements

The modular group $PSL_2(\mathbb{Z})$ acts on the hyperbolic half-space $H$ by $$h\cdot z=\frac{az+b}{cz+d},\;z\in H,\;h=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in PSL_2(\mathbb{Z})$$ with ...
0
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1answer
99 views

Convergent's numerators of the continued fraction for $\pi$

Call $C_{\pi} = \{ 1,3,22,333,355,… \}$, it´s the sequence of the numerators of convergents of the continued fraction for $\pi$, its OEIS' A002485, http://oeis.org/A002485. Let $n \in C_{\pi}$, such ...
3
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2answers
63 views

Continued fraction and classification of real numbers.

I would be grateful if anyone can tell if there are any methods to classify real numbers using continued fraction. eg: Suppose $[a_0;a_1,a_2,\ldots,a_n]$ is the representation of some real number ...
9
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3answers
173 views

Generalizations of $\sum_{m=3n+2}^{\infty}\phi^m=\phi^{3n}$ and $\sum_{m=13n+1}^{\infty}(\sqrt2-1)^m=\dfrac{(\sqrt2-1)^{13n}}{\sqrt2}$

I noticed that the following identies hold with the help of wolfram alpha and oeis. I'm sure they're well-known, but I'd like to know how they generalize. ...
16
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1answer
357 views

An infinite series plus a continued fraction by Ramanujan

Here is a famous problem posed by Ramanujan Show that $$\left(1 + \frac{1}{1\cdot 3} + \frac{1}{1\cdot 3\cdot 5} + \cdots\right) + ...
0
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1answer
15 views

A weak inequality than Hurwitz

How can i prove that among any two consecuent convergents to x, al least one of the satisfy $|x-h_{n}/k_{n}|$ $< 1/2k_{n}^2$ I know, by the Thoerem of Hurwitz, that among any three consecutive ...
0
votes
0answers
29 views

About an rational aproximation to an irrational

How to show that if $x$ is an irrational number, then $x= a_{0} + \sum_{n=0}^{\infty} \frac{(-1)^n}{k_{n}k_{n+1}}$ where the $k_{n}$ are the denominators of the $n$th convergents to $x$? Maybe a ...
7
votes
1answer
257 views

Chinese estimate for $\pi$. Were they lucky?

The famous chinese estimate $\pi\approx\frac{355}{113}$ is good. I think that is too good. As a continued fraction: $$\pi=[3:7,15,1,292,\ldots]$$ That $292$ is a bit too big. Is there a reason for a ...
2
votes
1answer
31 views

How to find the terms of the continued fraction representation for $e^\pi$

The question is - Find the first ten terms of the continued fraction representation for $e^ \pi $
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1answer
84 views

Summing up numbers from the continued fraction of $e ^ \pi$ and $\pi ^e$

I don't remember it well ,but it was around 5-6 years ago , I was 8 and I had found this new interest - continued fractions .I used to play with their terms sum them up and thought of getting ...
1
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1answer
100 views

Proof of continued fraction convergence theorem

How does one prove that $$|\alpha - \frac{p_n}{q_n}| < |\alpha - \frac{p_m}{q_m}|$$ for all $n>m$? I know that the left side is less than $\frac{1}{2q_n^2}$ and the right side is less than ...
0
votes
1answer
99 views

Continued fraction proof

I am really confused about the proof of this theorem: For any continued fraction, $$q_n\alpha - p_n = \frac{(-1)^n}{\alpha_{n+1}q_n + q_{n-1}}$$ I got the base induction case for $n=0$ but I can't ...
3
votes
1answer
107 views

Rational aproximations of golden ratio

I read a blogpost that mentions that for golden ratio, the sets of best rational approximations of the first kind and the second kind are the same. Is this true? If so, why? Are there other numbers ...
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0answers
32 views

PDF describing nth term in continued fraction

For a real number r chosen uniformly at random in the range (0,1), what's the marginal Probability Density Function that describes the nth term in the continued fraction representation of r? What ...
0
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1answer
40 views

How to write continued fraction as polynomial?

I have \begin{align} r(x)= 1 + \frac{x}{\frac{1}{2}+\frac{x-1}{-1+\frac{x+1}{1+\frac{x-1}{-1}}}} \end{align} for an interpolation problem, and I need to write $r(x)$ such that nominator and ...
3
votes
2answers
76 views

Evaluating $\frac{2}{3-\frac{2}{3-\frac{2}{3-\ldots} } }$

What is the value of the fraction $\frac{2}{3-\frac{2}{3-\frac{2}{3-\ldots} } } $? I let $x=\frac{2}{3-\frac{2}{3-\frac{2}{3-\ldots} } } $ and hence $x=\frac{2}{3-x} $ which gives $x=1$ or $x=2$. ...
1
vote
2answers
33 views

functions on a continued fraction expansion

Let $x$ be an irrational number with continued fraction expansion $[a_0;a_1,a_2,\ldots]$. Is there an $x$ and a non-identity function $f$ such that $f(x)=[f(a_0);f(a_1),f(a_2),\ldots]$. Given that I ...
4
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0answers
171 views

Cube root of two continued fraction

I know there is a nice way of getting the continued fraction expansion of quadratic irrationals mainly because they recur after a point, and if they recur after a point they are quadratic irrationals. ...
2
votes
2answers
171 views

Question in fraction (not simple )

I have a question and its answer but I don't know how can i solve $$\frac {37}{13} = 2+ \frac {1}{x+\frac{1}{5+\frac{1}{y}}} $$ the answer $ x =1, y=2$ Could any one explain how to solve this ?? ...
3
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0answers
76 views

Updated:Sum of entries in continued fraction of $\sqrt d$ and $\sqrt{d}-\lfloor \sqrt{d}\rfloor$ equals (divides) $d$.

(1)I noted as a joke in class, for $\sqrt{13}$ which has continued fraction expansion $[3;\overline{1,1,1,1,6}]$ that $3+1+1+1+1+6=13$. Another eg. $\sqrt{22}=[4;\overline{{1,2,4,2,1,8}}]$, as ...